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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fdvposle | Structured version Visualization version GIF version |
Description: Functions with a nonnegative derivative, i.e. monotonously growing functions, preserve ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
Ref | Expression |
---|---|
fdvposlt.d | β’ πΈ = (πΆ(,)π·) |
fdvposlt.a | β’ (π β π΄ β πΈ) |
fdvposlt.b | β’ (π β π΅ β πΈ) |
fdvposlt.f | β’ (π β πΉ:πΈβΆβ) |
fdvposlt.c | β’ (π β (β D πΉ) β (πΈβcnββ)) |
fdvposle.le | β’ (π β π΄ β€ π΅) |
fdvposle.1 | β’ ((π β§ π₯ β (π΄(,)π΅)) β 0 β€ ((β D πΉ)βπ₯)) |
Ref | Expression |
---|---|
fdvposle | β’ (π β (πΉβπ΄) β€ (πΉβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioossicc 13452 | . . . . . 6 β’ (π΄(,)π΅) β (π΄[,]π΅) | |
2 | 1 | a1i 11 | . . . . 5 β’ (π β (π΄(,)π΅) β (π΄[,]π΅)) |
3 | ioombl 25522 | . . . . . 6 β’ (π΄(,)π΅) β dom vol | |
4 | 3 | a1i 11 | . . . . 5 β’ (π β (π΄(,)π΅) β dom vol) |
5 | fdvposlt.c | . . . . . . . 8 β’ (π β (β D πΉ) β (πΈβcnββ)) | |
6 | cncff 24841 | . . . . . . . 8 β’ ((β D πΉ) β (πΈβcnββ) β (β D πΉ):πΈβΆβ) | |
7 | 5, 6 | syl 17 | . . . . . . 7 β’ (π β (β D πΉ):πΈβΆβ) |
8 | 7 | adantr 479 | . . . . . 6 β’ ((π β§ π₯ β (π΄[,]π΅)) β (β D πΉ):πΈβΆβ) |
9 | fdvposlt.d | . . . . . . . 8 β’ πΈ = (πΆ(,)π·) | |
10 | fdvposlt.a | . . . . . . . 8 β’ (π β π΄ β πΈ) | |
11 | fdvposlt.b | . . . . . . . 8 β’ (π β π΅ β πΈ) | |
12 | 9, 10, 11 | fct2relem 34270 | . . . . . . 7 β’ (π β (π΄[,]π΅) β πΈ) |
13 | 12 | sselda 3982 | . . . . . 6 β’ ((π β§ π₯ β (π΄[,]π΅)) β π₯ β πΈ) |
14 | 8, 13 | ffvelcdmd 7100 | . . . . 5 β’ ((π β§ π₯ β (π΄[,]π΅)) β ((β D πΉ)βπ₯) β β) |
15 | ioossre 13427 | . . . . . . . 8 β’ (πΆ(,)π·) β β | |
16 | 9, 15 | eqsstri 4016 | . . . . . . 7 β’ πΈ β β |
17 | 16, 10 | sselid 3980 | . . . . . 6 β’ (π β π΄ β β) |
18 | 16, 11 | sselid 3980 | . . . . . 6 β’ (π β π΅ β β) |
19 | ax-resscn 11205 | . . . . . . . 8 β’ β β β | |
20 | ssid 4004 | . . . . . . . 8 β’ β β β | |
21 | cncfss 24847 | . . . . . . . 8 β’ ((β β β β§ β β β) β ((π΄[,]π΅)βcnββ) β ((π΄[,]π΅)βcnββ)) | |
22 | 19, 20, 21 | mp2an 690 | . . . . . . 7 β’ ((π΄[,]π΅)βcnββ) β ((π΄[,]π΅)βcnββ) |
23 | 7, 12 | feqresmpt 6973 | . . . . . . . 8 β’ (π β ((β D πΉ) βΎ (π΄[,]π΅)) = (π₯ β (π΄[,]π΅) β¦ ((β D πΉ)βπ₯))) |
24 | rescncf 24845 | . . . . . . . . 9 β’ ((π΄[,]π΅) β πΈ β ((β D πΉ) β (πΈβcnββ) β ((β D πΉ) βΎ (π΄[,]π΅)) β ((π΄[,]π΅)βcnββ))) | |
25 | 12, 5, 24 | sylc 65 | . . . . . . . 8 β’ (π β ((β D πΉ) βΎ (π΄[,]π΅)) β ((π΄[,]π΅)βcnββ)) |
26 | 23, 25 | eqeltrrd 2830 | . . . . . . 7 β’ (π β (π₯ β (π΄[,]π΅) β¦ ((β D πΉ)βπ₯)) β ((π΄[,]π΅)βcnββ)) |
27 | 22, 26 | sselid 3980 | . . . . . 6 β’ (π β (π₯ β (π΄[,]π΅) β¦ ((β D πΉ)βπ₯)) β ((π΄[,]π΅)βcnββ)) |
28 | cniccibl 25798 | . . . . . 6 β’ ((π΄ β β β§ π΅ β β β§ (π₯ β (π΄[,]π΅) β¦ ((β D πΉ)βπ₯)) β ((π΄[,]π΅)βcnββ)) β (π₯ β (π΄[,]π΅) β¦ ((β D πΉ)βπ₯)) β πΏ1) | |
29 | 17, 18, 27, 28 | syl3anc 1368 | . . . . 5 β’ (π β (π₯ β (π΄[,]π΅) β¦ ((β D πΉ)βπ₯)) β πΏ1) |
30 | 2, 4, 14, 29 | iblss 25762 | . . . 4 β’ (π β (π₯ β (π΄(,)π΅) β¦ ((β D πΉ)βπ₯)) β πΏ1) |
31 | 7 | adantr 479 | . . . . 5 β’ ((π β§ π₯ β (π΄(,)π΅)) β (β D πΉ):πΈβΆβ) |
32 | 2 | sselda 3982 | . . . . . 6 β’ ((π β§ π₯ β (π΄(,)π΅)) β π₯ β (π΄[,]π΅)) |
33 | 32, 13 | syldan 589 | . . . . 5 β’ ((π β§ π₯ β (π΄(,)π΅)) β π₯ β πΈ) |
34 | 31, 33 | ffvelcdmd 7100 | . . . 4 β’ ((π β§ π₯ β (π΄(,)π΅)) β ((β D πΉ)βπ₯) β β) |
35 | fdvposle.1 | . . . 4 β’ ((π β§ π₯ β (π΄(,)π΅)) β 0 β€ ((β D πΉ)βπ₯)) | |
36 | 30, 34, 35 | itgge0 25768 | . . 3 β’ (π β 0 β€ β«(π΄(,)π΅)((β D πΉ)βπ₯) dπ₯) |
37 | fdvposle.le | . . . 4 β’ (π β π΄ β€ π΅) | |
38 | fdvposlt.f | . . . . 5 β’ (π β πΉ:πΈβΆβ) | |
39 | fss 6744 | . . . . 5 β’ ((πΉ:πΈβΆβ β§ β β β) β πΉ:πΈβΆβ) | |
40 | 38, 19, 39 | sylancl 584 | . . . 4 β’ (π β πΉ:πΈβΆβ) |
41 | cncfss 24847 | . . . . . 6 β’ ((β β β β§ β β β) β (πΈβcnββ) β (πΈβcnββ)) | |
42 | 19, 20, 41 | mp2an 690 | . . . . 5 β’ (πΈβcnββ) β (πΈβcnββ) |
43 | 42, 5 | sselid 3980 | . . . 4 β’ (π β (β D πΉ) β (πΈβcnββ)) |
44 | 9, 10, 11, 37, 40, 43 | ftc2re 34271 | . . 3 β’ (π β β«(π΄(,)π΅)((β D πΉ)βπ₯) dπ₯ = ((πΉβπ΅) β (πΉβπ΄))) |
45 | 36, 44 | breqtrd 5178 | . 2 β’ (π β 0 β€ ((πΉβπ΅) β (πΉβπ΄))) |
46 | 38, 11 | ffvelcdmd 7100 | . . 3 β’ (π β (πΉβπ΅) β β) |
47 | 38, 10 | ffvelcdmd 7100 | . . 3 β’ (π β (πΉβπ΄) β β) |
48 | 46, 47 | subge0d 11844 | . 2 β’ (π β (0 β€ ((πΉβπ΅) β (πΉβπ΄)) β (πΉβπ΄) β€ (πΉβπ΅))) |
49 | 45, 48 | mpbid 231 | 1 β’ (π β (πΉβπ΄) β€ (πΉβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3949 class class class wbr 5152 β¦ cmpt 5235 dom cdm 5682 βΎ cres 5684 βΆwf 6549 βcfv 6553 (class class class)co 7426 βcc 11146 βcr 11147 0cc0 11148 β€ cle 11289 β cmin 11484 (,)cioo 13366 [,]cicc 13369 βcnβccncf 24824 volcvol 25420 πΏ1cibl 25574 β«citg 25575 D cdv 25820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cc 10468 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-symdif 4245 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-ofr 7693 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-omul 8500 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-fi 9444 df-sup 9475 df-inf 9476 df-oi 9543 df-dju 9934 df-card 9972 df-acn 9975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-ioo 13370 df-ioc 13371 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-fl 13799 df-mod 13877 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-limsup 15457 df-clim 15474 df-rlim 15475 df-sum 15675 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-rest 17413 df-topn 17414 df-0g 17432 df-gsum 17433 df-topgen 17434 df-pt 17435 df-prds 17438 df-xrs 17493 df-qtop 17498 df-imas 17499 df-xps 17501 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-mulg 19038 df-cntz 19282 df-cmn 19751 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22824 df-topon 22841 df-topsp 22863 df-bases 22877 df-cld 22951 df-ntr 22952 df-cls 22953 df-nei 23030 df-lp 23068 df-perf 23069 df-cn 23159 df-cnp 23160 df-haus 23247 df-cmp 23319 df-tx 23494 df-hmeo 23687 df-fil 23778 df-fm 23870 df-flim 23871 df-flf 23872 df-xms 24254 df-ms 24255 df-tms 24256 df-cncf 24826 df-ovol 25421 df-vol 25422 df-mbf 25576 df-itg1 25577 df-itg2 25578 df-ibl 25579 df-itg 25580 df-0p 25627 df-limc 25823 df-dv 25824 |
This theorem is referenced by: fdvnegge 34275 |
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